Question

Some PLCs have PROPORTIONAL BAND % settings, whilst others have GAIN settings. If you calculate GAIN how would it be possible to convert to PROPORTIONAL BAND%? Please explain in detail.

Answer 1

The PB (Proportional Band) is the inverse of the Gain. One can write:
PB = 100 / Gain or

Gain = 100 / PB

where PB is in percent.

The higher gain the faster (and less stable) control.
The lower gain the slower (and more stable) control.

Conversion to proportional band that means:
The higher Pb the slower (and more stable) control.
The lower Pb the faster (and less stable) control.

Take, for example, a programmable level controller on a tank where we measure the level from bottom to top as 0 to 100%. We have a control valve on the outlet of the tank whose job is to maintain the level in the tank. The PB is defined as the range of level over which the control valve will go from fully closed to fully open. Suppose we decide that if the tank level should fall to 20% we want the control valve fully closed (0% open). Additionally, if the tank level rises to 60% full, we want the control valve to be fully open (100% open). If the tank level is between 20% and 60% we want the control valve to be open "in proportion" the the level. So if the tank level were to rise to 40% (half way up the PB), the control valve should be set to 50% open). This controller would have a PB of 40% (60% - 20%).

A proportional band of 250% is a bit tougher to visualize. Since the process, the tank in our case, can only go 100%, then a PB of 250% causes the valve to move only through a portion of its available stroke as the tank goes from empty to 100% full (assuming proportional only control - Integral and/or derivative action acts in addition to proportional).

Let's assume that we set the lower end of the proportional band at -75% and the upper end at 175%. The gives a PB of 250%. Physically it means that the level would have to fall to below (way below) the bottom of the tank to fully close the valve and above (way above) to fully open the valve. Since the level of our tank can't really go that far, it means that the valve will never fully close or fully open. This occasionally happens when the need for gradual valve movement as the process changes is more important than having a control valve that can close when the level gets low. Tuning is the process of finding the gain or PB that provides the optimal control valve response for controlling the process.

Many process controllers will also have integral and/or derivative actions that act in addition to the proportional control.

i.e.
Pb=25%, GAIN=100/25 = 4 = very reactive, typical level control, stable PV & linear relationship with output
Pb=250%, GAIN=100/250 = 0.4 = not very reactive, typical flow/pressure, or noisy PV, or non linear (usually high integral).